The velocity of an object, in feet per second, is given in the table.
a) Sketch a velocity-time graph from the table.
b) Give lower and upper estimates for distance traveled in the 5 seconds.
c) Estimate actual distance traveled.
d) Show a representation of the lower estimate in terms of a shaded region on the graph.
(1.4)(1) + (2.7)(1) + (3.5)(1) + (4.5)(1) + (5)(1)
= 17.1 ft.
(2.7)(1) + (3.5)(1) + (4.5)(1) + (5)(1) + (5.7)(1)
= 21.4 ft.
c) [17.1+ 21.4] / 2
= 19.25 ft.
d) The graph is monotonic increasing, so the lower estimate can be found by the left hand sum. This part is shaded in grey.
Area is width multiplied by the height. Integral is the signed area under a curve. If you look at the first example, from 0 to 3 the area you would get is 9/2. You would also get the same area from -3 to 0. But if we're talking about integrals, the integral from 0 to 3 is 9/2 and the integral from -3 to 0 is -9/2.
Notice that in the first and second examples, we're easily able to find the area because the functions are shaped nicely... into triangles and a half circle. In the third example however, we get a parabola... so to find the area, we'd have to find the different values at different intervals and add the values together as we've done on our previous questions.
After this, we programmed Riemann's sum (RSUM) into our calculators. As usual, we ran out of time soo if you guys missed the steps, just turn to your handy dandy textbooks and you'll find it all on page 168.
The HOMEWORK for tonight is Excercise 3.2 all ODD questions only. And tomorrow's scribe shall be MARK.